Cramer’s Rule Practice Test

Advanced Algebra Practice Test: ACT math skills.

Cramer’s Rule Practice Test

This test has 20 questions

Instant feedback · Worked explanations

Determinant Questions · High School Linear Algebra

Keep the columns in order. Replace only the one you need.

This free Cramer’s Rule Practice Test contains 20 multiple-choice questions and does not require registration. The questions are written for high school Advanced Algebra practice and focus on writing systems in standard form, building a coefficient determinant, replacing the correct column with constants, evaluating small determinants, dividing by the original determinant, recognizing when the rule cannot be used, and checking the resulting solution. Each question has four answer choices, one correct answer, and a detailed explanation that keeps the column-replacement logic visible from start to finish.

Cramer route

Arrangestandard form

Buildbase determinant

Replaceone column

Divideand verify

System

Base determinant

Column switch

Ratio

Check

Purpose

Cramer’s Rule turns a small linear system into determinant ratios

For two equations with two unknowns, the same coefficient determinant becomes the denominator of both answers.

Core idea

Numerator ticket

D_{x} or D_{y}

Replace the column belonging to the variable being found.

\div

Denominator ticket

D

Keep the original coefficient columns exactly as written.

Template

Read the general two-equation template by columns

The first coefficient column belongs to the first unknown, the second coefficient column belongs to the second unknown, and the constants form a separate column.

Column order

\begin{matrix} a x + b y & = & e \\ c x + d y & = & f \end{matrix}

First column

[\begin{matrix} a \\ c \end{matrix}]

Coefficients of the first unknown.

Second column

[\begin{matrix} b \\ d \end{matrix}]

Coefficients of the second unknown.

Constants column

[\begin{matrix} e \\ f \end{matrix}]

Values on the right side of the equations.

Build

Build the base determinant and two replacement determinants

Only one column changes at a time. The other coefficient column stays in its original position.

Three tickets

Original

D = \det ([\begin{matrix} a & b \\ c & d \end{matrix}])

No replacement. This determinant is the common denominator.

First unknown

D_{x} = \det ([\begin{matrix} e & b \\ f & d \end{matrix}])

The constants replace the first coefficient column.

Second unknown

D_{y} = \det ([\begin{matrix} a & e \\ c & f \end{matrix}])

The constants replace the second coefficient column.

Switch

Visualize replacement as a one-column switchyard

The constants travel into exactly one location while every non-replaced column stays on its track.

Visual map

Replacement is not insertion

The determinant must keep the same number of columns. Remove the chosen coefficient column and put the constants in that exact position.

Example

Worked example: begin with a system already in standard form

The coefficient columns and constants can be read directly because matching unknowns are aligned.

School example

\begin{matrix} 2 x + y & = & 7 \\ x - y & = & 2 \end{matrix}

The first coefficient column is made from the coefficients of

x

, the second from the coefficients of

y

, and the final column is made from the constants.

Base

Calculate the base determinant first

This early calculation tells you whether Cramer’s Rule can produce a unique solution.

Denominator

D = \det ([\begin{matrix} 2 & 1 \\ 1 & - 1 \end{matrix}]) = 2 (- 1) - 1 (1) = - 3

Order of products

Use main-diagonal product minus cross-diagonal product.

Eligibility check

The result is nonzero, so both determinant ratios are defined.

First swap

Replace the first column to build the first numerator

The constants move into the first column while the second coefficient column remains unchanged.

Find first unknown

D_{x} = \det ([\begin{matrix} 7 & 1 \\ 2 & - 1 \end{matrix}]) = 7 (- 1) - 1 (2) = - 9

A common wrong setup places the constants in the second column. That setup belongs to the other unknown.

Second swap

Restore the original matrix, then replace the second column

Each numerator starts from the original coefficient matrix, not from the previous replacement matrix.

Find second unknown

D_{y} = \det ([\begin{matrix} 2 & 7 \\ 1 & 2 \end{matrix}]) = 2 (2) - 7 (1) = - 3

The original first column returns before the constants replace the second column.

Ratios

Divide each replacement determinant by the base determinant

A negative numerator divided by a negative denominator gives a positive value in both calculations.

Solution

First unknown

x = \frac{D_{x}}{D} = \frac{- 9}{- 3} = 3

Second unknown

y = \frac{D_{y}}{D} = \frac{- 3}{- 3} = 1

Gate

Every answer passes through the same denominator gate

The numerator changes with the requested unknown, but the base determinant never changes.

Structure

One invariant, two numerators

If the denominator changes between the two ratios, the determinant setup has been copied incorrectly.

Verify

Substitution is the fastest final check

Put the ordered pair into both original equations. A correct solution must satisfy both, not just one.

Two checks

First equation

2 (3) + 1 = 7

The left side matches the first constant.

Second equation

3 - 1 = 2

The left side also matches the second constant.

Arrange

Rearrange equations before reading coefficient columns

Unknowns must appear in the same order, and constants must be isolated on the same side.

Preparation

Given form

\begin{matrix} y & = & 5 - x \\ 2 x - y & = & 1 \end{matrix}

The first equation is not yet aligned with the second.

Standard form

\begin{matrix} x + y & = & 5 \\ 2 x - y & = & 1 \end{matrix}

Now every column has one clear meaning.

D = - 3, D_{x} = - 6, D_{y} = - 9 \Rightarrow x = 2, y = 3

Decision

A zero base determinant closes the Cramer ratio route

Division by zero is undefined. The system may have no solution or infinitely many solutions, so another method is needed to classify it.

Stop condition

Zero does not name the solution type

It only says that Cramer’s Rule cannot divide to produce one unique ordered pair.

Classify

Two systems can share a zero determinant but behave differently

Compare the equations after the base determinant reaches zero.

No division

Same line

\begin{matrix} x + y & = & 4 \\ 2 x + 2 y & = & 8 \end{matrix}

The second equation is twice the first, so infinitely many ordered pairs work.

Parallel lines

\begin{matrix} x + y & = & 4 \\ 2 x + 2 y & = & 9 \end{matrix}

The coefficient rows are proportional but the constants are not, so there is no solution.

Parameter

A parameter question often asks when the Cramer route is open

Build the base determinant as an expression and find when it is nonzero.

Unique solution

System

\begin{matrix} k x + y & = & 4 \\ 2 x + 2 y & = & 6 \end{matrix}

Condition

D = 2 k - 2 = 2 (k - 1)

D \neq 0 \Rightarrow k \neq 1

The system has one unique solution for every allowed parameter value shown.

Extend

The same column rule extends to three unknowns

Use a three-by-three coefficient determinant and create one replacement determinant for each unknown. The organizational rule does not change.

Three variables

\begin{matrix} x + y + z & = & 6 \\ x - y + z & = & 2 \\ 2 x + y - z & = & 1 \end{matrix}

Determinant values

D = 6, D_{x} = 6, D_{y} = 12, D_{z} = 18

Ratios

x = \frac{6}{6}, y = \frac{12}{6}, z = \frac{18}{6}

Solution

(x, y, z) = (1, 2, 3)

Substitution confirms all three equations.

Skills

Skills Covered

These medium-level high school Advanced Algebra questions combine system organization, determinant accuracy, and careful interpretation.

Coverage

Standard form

Align unknowns and constants before copying coefficients.

Coefficient matrices

Preserve row order and variable-column order.

Column replacement

Replace only the column associated with the requested unknown.

Determinant arithmetic

Use diagonal products and signs accurately.

Eligibility

Recognize that a nonzero base determinant is required.

Verification

Substitute the final values into every original equation.

Approach

How to Approach the Test

Treat each determinant as a labeled ticket so that a correct calculation is never attached to the wrong variable.

Route plan

01StandardizePut equations in matching variable order with constants on the right.

02Build the baseCopy only coefficients and calculate the original determinant first.

03Check the gateContinue with Cramer’s Rule only when the base determinant is nonzero.

04Label replacementsReplace the first column for the first unknown and the second for the second unknown.

05Divide and verifyUse the same base denominator, simplify signs, and test the ordered pair.

Mistakes

Common Mistakes

Most distractors come from a correct determinant rule applied to the wrong column arrangement.

Avoid

Reading before rearranging

Unaligned equations produce coefficient columns with the wrong meaning.

Using constants in the base

The original determinant contains coefficients only.

Appending a third column

Constants replace a column; they are not added beside the matrix.

Replacing both columns

Each numerator changes exactly one column.

Keeping the previous swap

Restore the original coefficient matrix before building the next numerator.

Reversing diagonal products

Use main product minus cross product consistently.

Dividing in the wrong direction

The replacement determinant is the numerator and the base determinant is the denominator.

Dividing by zero

A zero base determinant means the Cramer ratios are unavailable.

Stopping after one check

A proposed ordered pair must satisfy every equation in the system.

Audit

Final route audit

Confirm the track, the switch, the ratio, and the destination.

Arrange → Base → Replace → Divide → Check

Are the equations in matching standard form?Variables must use the same column order.

Does the base determinant contain coefficients only?Constants do not enter until a replacement is built.

Is the base determinant nonzero?This is required before any Cramer ratio is used.

Was exactly the requested variable column replaced?Keep every other coefficient column unchanged.

Is the replacement value above the base value?Protect negative values with clear fraction grouping.

Does the final solution satisfy every original equation?Substitution catches column and sign errors quickly.

Use this free 20-question practice test for high school Advanced Algebra review, placement preparation, or classroom practice. You can retake the test without creating an account. The examples in this review block are illustrative and are not copies of the test questions.